Ultraspherical polynomials

Gegenbauer polynomials

Orthogonal polynomials $P _ {n} ( x, \lambda )$ on the interval $[ - 1 , 1 ]$ with the weight function $h ( x) = ( 1 - x ^ {2} ) ^ {\lambda - 1 / 2 }$; a particular case of the Jacobi polynomials for $\alpha = \beta = \lambda - 1 / 2$( $\lambda > - 1 / 2$); the Legendre polynomials $P _ {n} ( x)$ are a particular case of the ultraspherical polynomials: $P _ {n} ( x) = P _ {n} ( x , 1 / 2 )$.

For ultraspherical polynomials one has the standardization

$$P _ {n} ( x , \lambda ) \equiv \ C _ {n} ^ {( \lambda ) } ( x) =$$

$$= \ \frac{( - 2 ) ^ {n} }{n!} \frac{\Gamma ( n + \lambda ) \Gamma ( n + 2 \lambda ) }{\Gamma ( \lambda ) \Gamma ( 2 n + 2 \lambda ) } ( 1 - x ^ {2} ) ^ {- \lambda + 1 / 2 } \times$$

$$\times \frac{d ^ {n} }{d x ^ {n} } [ ( 1 - x ^ {2} ) ^ {n + \lambda - 1 / 2 } ]$$

and the representation

$$C _ {n} ^ {( \lambda ) } ( x) = \ \sum _ { k= 0} ^ { [ n / 2 ] } ( - 1 ) ^ {k} \frac{\Gamma ( n - k + \lambda ) }{\Gamma ( \lambda ) k ! ( n - 2 k ) ! } ( 2 x ) ^ {n-} 2k .$$

The ultraspherical polynomials are the coefficients of the power series expansion of the generating function

$$\frac{1}{( 1 - 2 x w + w ^ {2} ) ^ \lambda } = \ \sum _ { n= 0} ^ \infty C _ {n} ^ {( \lambda ) } ( x) w ^ {n} .$$

The ultraspherical polynomial $C _ {n} ^ {( \lambda ) } ( x)$ satisfies the differential equation

$$( 1 - x ^ {2} ) y ^ {\prime\prime} - ( 2 \lambda + 1 ) x y ^ \prime + n ( n + 2 \lambda ) y = 0 .$$

More commonly used are the formulas

$$( n + 1 ) C _ {n+1} ^ {( \lambda ) } ( x) = \ 2 ( n + \lambda ) x C _ {n} ^ {( \lambda ) } ( x) - ( n + 2 \lambda - 1 ) C _ {n-1} ^ {( \lambda ) } ( x) ,$$

$$C _ {n} ^ {( \lambda ) } ( - x ) = ( - 1 ) ^ {n} C _ {n} ^ {( \lambda ) } ( x) ,$$

$$\frac{d}{dx} [ C _ {n} ^ {( \lambda ) } ( x) ] = 2 \lambda C _ {n-1} ^ {( \lambda + 1) } ( x) ,$$

$$C _ {n} ^ {( \lambda ) } ( \pm 1 ) = ( \pm 1 ) ^ {n} \frac{2 \lambda ( 2 \lambda + 1 ) \dots ( 2 \lambda + n - 1 ) }{n!\ } =$$

$$= \ ( \pm 1 ) ^ {n} \left ( \begin{array}{c} n + 2 \lambda - 1 \\ n \end{array} \right ) .$$

For references see Orthogonal polynomials.

Comments

See Spherical harmonics for a group-theoretic interpretation. Ultraspherical polynomials are also connected with Jacobi polynomials by the quadratic transformations

$$C _ {2n} ^ {( \lambda ) } ( x) = \ \textrm{ const } P _ {n} ^ {( \lambda - 1/2 , - 1/2) } ( 2x ^ {2} - 1) ,$$

$$C _ {2n+ 1 } ^ {( \lambda ) } ( x) = \textrm{ const } x P _ {n} ^ {( \lambda - 1/2, 1/2) } ( 2x ^ {2} - 1) .$$

See [a1] for $q$-ultraspherical polynomials.

References

[a1]	R.A. Askey, M.E.H. Ismail, "A generalization of ultraspherical polynomials" P. Erdös (ed.) , Studies in Pure Mathematics to the Memory of Paul Turán , Birkhäuser (1983) pp. 55–78